Emergent coherent modes in nonlinear magnonic waveguides detected at ultrahigh frequency resolution

Nonlinearity of dynamic systems plays a key role in neuromorphic computing, which is expected to reduce the ever-increasing power consumption of machine learning and artificial intelligence applications. For spin waves (magnons), nonlinearity combined with phase coherence is the basis of phenomena like Bose–Einstein condensation, frequency combs, and pattern recognition in neuromorphic computing. Yet, the broadband electrical detection of these phenomena with high-frequency resolution remains a challenge. Here, we demonstrate the generation and detection of phase-coherent nonlinear magnons in an all-electrical GHz probe station based on coplanar waveguides connected to a vector network analyzer which we operate in a frequency-offset mode. Making use of an unprecedented frequency resolution, we resolve the nonlocal emergence of a fine structure of propagating nonlinear magnons, which sensitively depends on both power and a magnetic field. These magnons are shown to maintain coherency with the microwave source while propagating over macroscopic distances. We propose a multi-band four-magnon scattering scheme that is in agreement with the field-dependent characteristics of coherent nonlocal signals in the nonlinear excitation regime. Our findings are key to enable the seamless integration of nonlinear magnon processes into high-speed microwave electronics and to advance phase-encoded information processing in magnonic neuronal networks.


Field dependence at different power levels
We present the dependence of the field on various power levels in Fig. S1.At powers exceeding 0.05 mW, we observe an exceptionally narrow f p /2 peak.As the field increases, this peak begins to split.The splitting progressively shifts to lower fields with an increase in power.This trend follows a field-power relationship of −0.18 mT/mW, consistent with the observations noted in Fig. 2 of the main text.The converging branch observed in CPW2 demonstrates a similar downward shift with an increase in power.
We note that the amount of frequency splitting decreases with increasing power.To quantitatively model the field dependence at different powers, we considered the nonlinear magnon frequency shift, which was extensively studied in earlier works [1,2].We use the following empirical form: ∆f = P [a(µ 0 H) 2 + b(µ 0 H) + c], where P is the injection power and µ 0 H is the magnetic field.The coefficients a = −2.8114× 10 −4 GHz/(mT 2 • mW), b = 8.1363 × 10 −3 GHz/(mT • mW), and c = 5 × 10 −4 GHz/mW were determined to fit the measured field dependence at different powers.This model predicts frequency shifts of on the order of tens of MHz within a field range between 28 and 30 mT when P increases from 1 mW to 4 mW.
We compare the measured spectral evolution with our calculation based on the power dependence, which shows a reasonable agreement across the investigated power range.The excitation frequency was fp = 4.25 GHz.The nonlocal magnon branch starts to emerge above 1 mW as seen in the CPW2 spectra.The field for splitting shifts down with increasing power, consistent with the shift shown in Fig. 2 of the main text.Correspondingly the second converging point in the CPW2 spectra also shifts down with power.Dashed blue and red lines are the calculated field dependence for the counter-and co-propagating processes, respectively.

S-parameter measurement
The S-parameter measurement was taken to extract the basic magnetic properties of our film.Figure S2 shows the measured S21 amplitude.The maximum in the color plot is best-fitted with the well-known backward volume mode with k y = π/w YIG and k x = 0.5 rad/µm with the following parameters : γ/(2π) = 28 GHz/T, µ 0 M s = 0.176 T, and D ex = 5.4 × 10 −17 T/m 2 .The vertical green line shows that 28.7 mT is required to place the resonance at 2.125 GHz, which agrees reasonably well with the measured field value in the frequency offset measurement.The extracted parameters were used to calculate the magnon dispersion curves in the discussion section.

Detection without establishing phase synchronization
Stimulated by the phase sensitive detection of nonlinear waves, we further explored the influence of phase synchronization on the measured spectrum at CPW2 by replacing VNA port 1 with an external microwave generator (compare Fig. S3a (VNA only, 10 average) with Fig. S3b (10 average) and Fig. S3c (single shot)).
These studies were conducted under very similar environmental conditions, without remounting the sample between measurements.Nonetheless, these tests were carried out subsequent to the remounting of the sample following the initial experiments discussed in the main text, which may account for some discrepancies in the power evolution spectrum.Despite these slight variations, the side peak structures and half-frequency peaks are evident in all three measurement configurations.A substantial intensity fluctuation is observed in the single shot data from the external generator, as depicted in Fig. S3c.This fluctuation ranges from baseline noise levels to peak intensities due to phase randomness.Such variability is significantly subdued in the 10-times averaged data (Fig. S3b), resulting in a comparative reduction of peak intensity against the spectrum presented in Fig. S3a.This highlights the superiority of the phase-stable, all-electrical approach introduced in this work over traditional scalar detection methods.

Power dependent spectra over an extended field range
In Fig. 2 of the main text, we showed the evolution of spectra over the 1.2 mT field range, where the copropagating magnons are collected nonlocally in the CPW2 spectrum.Here we show the measurements over an extended field range of 27-29.1 mT.It should be noted that, although the same sample was used for these measurements, it was dismounted and remounted in the process.We attribute the observed variations in the low-field frequency structure to this procedure, which indicates the sensitivity of the system to the sample's precise orientation with respect to the external magnetic field.Nonetheless, the prominent branch unique to the CPW2 spectrum remains distinctly observable.
Fig. S4 Power dependent spectra over an extended field range.Power evolution spectra over an extended field range from a CPW1 and b CPW2.Below 28 mT the sharp fp/2 peak is still observable.More complicated side peak features develop at lower fields.Despite minor differences in low field characteristics, we observe consistently similar power evolution spectra from CPW1 and CPW2 spectra including the emerging branch in CPW2 attributed to the co-propagating magnons.

Linewidth of f p /2 peak
Here we provide a detailed view of the f p /2 peak, characterized by an exceptionally narrow linewidth on the order of 10 kHz.This narrow linewidth is indicative of the highly selective nature of the parametric pumping process and is limited by the instrumental bandwidth.Above 0.1 mW, the peak grows nearlinearly with power and decreases at the onset of frequency splitting, which occurs at approximately 1.45 mW.

Power dependent linewidth analysis
To analyze the evolution of linewidths with power, we performed Lorentzian fits to the spectra in Fig. 1f of the main text.We extracted peak positions (Fig. S6a) and Full Widths at Half Maximum (FWHM) (Fig. S6b).FWHM values reside between about 0.001 MHz and 10 MHz depending on the mode and power level.At low (high) power, the mode f p /2 has the relatively smallest (largest) linewidths which differ clearly from the linewidth ∆f expected from the Gilbert damping indicated by the horizontal line in Fig. S6b.The value of ∆f = 0.425 MHz is evaluated for a realistic damping parameter α = 1 × 10 −4 for thin YIG [3] according to ∆f = 2αf with f = 2.125 GHz.The extremely narrow linewidth for f p /2 at power levels below 1.3 mW suggests a distinct mechanism different from Gilbert damping.Here, we observe that the transverse dynamic magnetization component oscillates at f p /2 when the longitudinal component oscillates at f p .This is a coherently driven magnetic oscillation without a magnon relaxation process, allowing only a single frequency at f p /2 that is not governed by Gilbert damping.Above 2.3 mW, the linewidths of modes with f ̸ = f p /2 are on the order of sub-MHz and much larger than the linewidth of mode f p /2 at small power levels.This difference stems from different scattering processes.
At high powers, the 4-magnon scattering processes set in.They allow for more magnon pairs to satisfy the energy and momentum conservation rules as stated in Eq. 2 of the main text (f 1 = f p /2 − δ and f 2 = f p /2 + δ).These processes are different from Gilbert damping and determined by the inhomogeneity which contributes to the broadening of dispersion, increasing the number of possible magnon pairs in frequency space.

Detailed magnon spectrum at low field
At a magnetic field of 28 mT, multiple discrete side peaks become evident.A closer look of the spectrum is presented in Fig. S7a.Around 1 mW, we observe a variable frequency spacing, reaching a maximum of approximately 1.9 MHz (indicated by the green line in Fig. S7b).As power increases towards 2 mW, this spacing contracts to 1.1 MHz, with additional peaks manifesting at integer multiples of this spacing.
Further power increase results in a convergence at the half-frequency.Although we suspect that this phenomenon originates from the scattering among higher-order quantized modes along the y direction, a theoretical model that reproduces this behavior has yet to be developed.

Fig
Fig. S1 Field dependent spectra with various excitation powers.a CPW1 and b CPW2 signals detected by the VNA for nonlinear spin waves at different power levels as a function of in-plane field applied perpendicular to the CPWs.The excitation frequency was fp = 4.25 GHz.The nonlocal magnon branch starts to emerge above 1 mW as seen in the CPW2 spectra.The field for splitting shifts down with increasing power, consistent with the shift shown in Fig.2of the main text.Correspondingly the second converging point in the CPW2 spectra also shifts down with power.Dashed blue and red lines are the calculated field dependence for the counter-and co-propagating processes, respectively.

Fig. S2 S
Fig. S2 S parameter measurement.Field dependent transmission signal intensity (S21).The dashed grey line represents the calculated field dependence with the given material parameters for spin waves propagating with kx = 0.5 rad/µm.To excite fp/2 = 2.125 GHz, the required magnetic field is 28.7 mT, close to the field value in the frequency offset measurement.

Fig. S3
Fig. S3 Spectra comparison with and without phase synchronization.Transmission spectra comparison from Receiver B at VNA Port 2 a using input from VNA port 1 (averaged over 10 measurements).b using an external microwave generator (10 averages).c with single shot detection using the external generator.Despite the external generator's lack of phase synchronization with the VNA Receiver B, consistent spectral features are noted.However, significant signal fluctuations due to non-synchronized phases are evident in c, reduced by averaging in b.This averaging results in lower intensity compared to the VNA-only setup (a).The bottom panel linecut profiles at 5.6 mW highlight the signal-to-noise ratio variations.

Fig. S5
Fig. S5 Detailed view of the fp/2 peak.Linewidth of fp/2 peak at 28.6 mT a The half-frequency peak spans over only a few frequency points in the semi-log plot indicating an exceptionally narrow linewidth of about 10 kHz.b Power dependence shows that the peak exhibits near-linear growth with power until the frequency splitting point, beyond which its intensity rapidly decreases.

Fig. S6
Fig. S6 Power dependence of peak frequencies and their linewidths.a Peak frequencies above and below fp/2 and b their linewidths (FWHMs) including mode at fp/2 extracted from the spectra in Fig. 1f.The straight line in b indicates the linewidth expected from Gilbert damping with α = 1 × 10 −4 .The grey shaded area indicates the power range where multiple peaks overlap, preventing fitting with a single Lorentzian function.

Fig. S7
Fig. S7 Formation of the magnon frequency comb-like feature at low field.a Measured evolution of parametrically excited spectrum as a function of power at CPW1.Magnon frequency spacing sensitively changes with source power at 28 mT.b Color-coded horizontal linecuts are taken at 1.15 and 2.35 mW for the green and purple dashed lines in a.The frequency spacings are equidistant and reduces by almost a factor of 2 when increasing power twice.